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int(0)^(pi//2)(cosx)/((1+sin^(2)x))dx=?...

`int_(0)^(pi//2)(cosx)/((1+sin^(2)x))dx=?`

Text Solution

Verified by Experts

Let `I=int_(0)^(pi//2)(cosx)/(1+sin^(2)x)dx`
Put `sin x=t" "therefore" "cos x dx =dt`
When `x=0, t = sin 0 =0." When "x=(pi)/(2),t=sin.(pi)/(2)=1`
`therefore I=int_(0)^(1)(1)/(1+t^(2))dt=[tan^(-)t]_(0)^(1)`
`=tan^(-1)1- tan^(-1)0`
`=(pi)/(4)-0=(pi)/(4).`
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